The fundamental limits of single-anchor multi-antenna positioning are investigated. Exploiting the structure of the multiple input-multiple output-orthogonal frequency division multiplexing (MIMO-OFDM) channel at millimeter-wave frequencies, we present geometrically intuitive asymptotic expressions for the Fisher information on position, orientation and velocity for large bandwidth and number of antennas. The effects of synchronization errors and mobility are studied and it is shown that non-line-of-sight (NLOS) paths can be used to estimate the synchronization error and drastically improve the positioning performance. We also find that, in the presence of line-of-sight (LOS), mobility has a small impact on the achievable positioning accuracy, but in the NLOS-only scenario it can significantly improve the achievable performance, depending on the variance of the synchronization error. Finally, considering a communication system with device-specific transmission and reception constraints, we compare the positioning accuracy between the downlink and the uplink and show that they are equivalent under the same signal-to-noise ratio (SNR).